$L^{-1}\bigg[\frac{s + 2}{s^2 - 4s + 4 + 9}\bigg] = \bigg[\frac{s + 2}{(s - 2)^2 + 9}\bigg]$ $\therefore$ Adjusting -2 & 2 in numerator $$= L^{-1}\bigg[\frac{2 - 2 + 4}{(s - 2)^2 + 3^2}\bigg]$$

According to first shifting principle

$$ = e^{2t}L^{-1}\bigg[\frac{s + 4}{s^2 + 3^2}\bigg]$$

$$ = e^{2t}L^{-1}\bigg[\frac{s}{s^2 + 3^2} + \frac{4}{s^2 + 3^2}\bigg]$$

$$ = e^{2t}\bigg[cos3t + \frac{4}{3}sin3t\bigg]$$